Question

If the slope of one of the lines given by $a{x}^2+2hxy+b{y}^2=0$, be the square of the slope of the other, then $\frac{a+b}{h}+\frac{8h^2}{ab}$ is

Answer 1

Let $m$ and $m^2$ be the slopes of the lines represented by $a{x}^2+2hxy+b{y}^2=0$. Then,
$m+m^2=-\frac{2h}{b}$
and $m{m}^2=\frac{a}{b}=m^3$
$\Rightarrow (m+m^2{)}^3={(-\frac{2h}{b})}^3$
$\Rightarrow m^3+m^6+3m{m}^2(m+m^2)=(-\frac{8h^3}{b^3})$
$\Rightarrow \frac{a}{b^2}(a+b)+\frac{8h^3}{b^3}=\frac{6ah}{b^2}$
$\Rightarrow \frac{a+b}{h}+\frac{8h^2}{ab}=6$